ITP001 Axioms: ITP003^7.ax
%------------------------------------------------------------------------------
% File : ITP003^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : bool.ax [Gau19]
% : HL4003^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 223 ( 70 unt; 32 typ; 0 def)
% Number of atoms : 387 ( 140 equ; 45 cnn)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 1278 ( 45 ~; 61 |; 126 &; 779 @)
% ( 118 <=>; 149 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg; 779 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 210 ( 210 >; 0 *; 0 +; 0 <<)
% Number of symbols : 32 ( 30 usr; 3 con; 0-5 aty)
% Number of variables : 751 ( 90 ^ 577 !; 56 ?; 751 :)
% ( 28 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Ebool_2Eitself,type,
tyop_2Ebool_2Eitself: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Emin_2Eind,type,
tyop_2Emin_2Eind: $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Ebool_2EARB,type,
c_2Ebool_2EARB:
!>[A_27a: $tType] : A_27a ).
thf(c_2Ebool_2EBOUNDED,type,
c_2Ebool_2EBOUNDED: $o > $o ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Ebool_2EDATATYPE,type,
c_2Ebool_2EDATATYPE:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ELET,type,
c_2Ebool_2ELET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ebool_2EONE__ONE,type,
c_2Ebool_2EONE__ONE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > $o ) ).
thf(c_2Ebool_2EONTO,type,
c_2Ebool_2EONTO:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > $o ) ).
thf(c_2Ebool_2ERES__ABSTRACT,type,
c_2Ebool_2ERES__ABSTRACT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ebool_2ERES__EXISTS,type,
c_2Ebool_2ERES__EXISTS:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ERES__EXISTS__UNIQUE,type,
c_2Ebool_2ERES__EXISTS__UNIQUE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ERES__FORALL,type,
c_2Ebool_2ERES__FORALL:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ERES__SELECT,type,
c_2Ebool_2ERES__SELECT:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a ) ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ebool_2Eitself__case,type,
c_2Ebool_2Eitself__case:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Ebool_2Eitself @ A_27a ) > A_27b > A_27b ) ).
thf(c_2Ebool_2Eliteral__case,type,
c_2Ebool_2Eliteral__case:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ebool_2Ethe__value,type,
c_2Ebool_2Ethe__value:
!>[A_27a: $tType] : ( tyop_2Ebool_2Eitself @ A_27a ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Ebool_2EBOOL__CASES__AX,axiom,
! [V0t: $o] :
( ( V0t = c_2Ebool_2ET )
| ( V0t = c_2Ebool_2EF ) ) ).
thf(thm_2Ebool_2EETA__AX,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27a > A_27b] :
( ( ^ [V1x: A_27a] : ( V0t @ V1x ) )
= V0t ) ).
thf(thm_2Ebool_2ESELECT__AX,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1x: A_27a] :
( ( V0P @ V1x )
=> ( V0P @ ( c_2Emin_2E_40 @ A_27a @ V0P ) ) ) ).
thf(thm_2Ebool_2EINFINITY__AX,axiom,
? [V0f: tyop_2Emin_2Eind > tyop_2Emin_2Eind] :
( ( c_2Ebool_2EONE__ONE @ tyop_2Emin_2Eind @ tyop_2Emin_2Eind @ V0f )
& ( (~) @ ( c_2Ebool_2EONTO @ tyop_2Emin_2Eind @ tyop_2Emin_2Eind @ V0f ) ) ) ).
thf(thm_2Ebool_2ET__DEF,axiom,
( c_2Ebool_2ET
<=> ( ( ^ [V0x: $o] : V0x )
= ( ^ [V1x: $o] : V1x ) ) ) ).
thf(thm_2Ebool_2EFORALL__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Ebool_2E_21 @ A_27a )
= ( ^ [V0P: A_27a > $o] :
( c_2Emin_2E_3D @ ( A_27a > $o ) @ V0P
@ ^ [V1x: A_27a] : c_2Ebool_2ET ) ) ) ).
thf(thm_2Ebool_2EEXISTS__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Ebool_2E_3F @ A_27a )
= ( ^ [V0P: A_27a > $o] : ( V0P @ ( c_2Emin_2E_40 @ A_27a @ V0P ) ) ) ) ).
thf(thm_2Ebool_2EAND__DEF,axiom,
( c_2Ebool_2E_2F_5C
= ( ^ [V0t1: $o,V1t2: $o] :
( c_2Ebool_2E_21 @ $o
@ ^ [V2t: $o] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D_3D_3E @ V0t1 @ ( c_2Emin_2E_3D_3D_3E @ V1t2 @ V2t ) ) @ V2t ) ) ) ) ).
thf(thm_2Ebool_2EOR__DEF,axiom,
( c_2Ebool_2E_5C_2F
= ( ^ [V0t1: $o,V1t2: $o] :
( c_2Ebool_2E_21 @ $o
@ ^ [V2t: $o] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D_3D_3E @ V0t1 @ V2t ) @ ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D_3D_3E @ V1t2 @ V2t ) @ V2t ) ) ) ) ) ).
thf(thm_2Ebool_2EF__DEF,axiom,
( c_2Ebool_2EF
<=> ! [V0t: $o] : V0t ) ).
thf(thm_2Ebool_2ENOT__DEF,axiom,
( c_2Ebool_2E_7E
= ( ^ [V0t: $o] : ( c_2Emin_2E_3D_3D_3E @ V0t @ c_2Ebool_2EF ) ) ) ).
thf(thm_2Ebool_2EEXISTS__UNIQUE__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Ebool_2E_3F_21 @ A_27a )
= ( ^ [V0P: A_27a > $o] :
( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_3F @ A_27a @ V0P )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V1x: A_27a] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V2y: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( V0P @ V1x ) @ ( V0P @ V2y ) ) @ ( c_2Emin_2E_3D @ A_27a @ V1x @ V2y ) ) ) ) ) ) ) ).
thf(thm_2Ebool_2ELET__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Ebool_2ELET @ A_27a @ A_27b )
= ( ^ [V0f: A_27a > A_27b,V1x: A_27a] : ( V0f @ V1x ) ) ) ).
thf(thm_2Ebool_2ECOND__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Ebool_2ECOND @ A_27a )
= ( ^ [V0t: $o,V1t1: A_27a,V2t2: A_27a] :
( c_2Emin_2E_40 @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D @ $o @ V0t @ c_2Ebool_2ET ) @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V1t1 ) ) @ ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D @ $o @ V0t @ c_2Ebool_2EF ) @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V2t2 ) ) ) ) ) ) ).
thf(thm_2Ebool_2EONE__ONE__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Ebool_2EONE__ONE @ A_27a @ A_27b )
= ( ^ [V0f: A_27a > A_27b] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V1x1: A_27a] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V2x2: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D @ A_27b @ ( V0f @ V1x1 ) @ ( V0f @ V2x2 ) ) @ ( c_2Emin_2E_3D @ A_27a @ V1x1 @ V2x2 ) ) ) ) ) ) ).
thf(thm_2Ebool_2EONTO__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Ebool_2EONTO @ A_27a @ A_27b )
= ( ^ [V0f: A_27a > A_27b] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V1y: A_27b] :
( c_2Ebool_2E_3F @ A_27a
@ ^ [V2x: A_27a] : ( c_2Emin_2E_3D @ A_27b @ V1y @ ( V0f @ V2x ) ) ) ) ) ) ).
thf(thm_2Ebool_2ETYPE__DEFINITION,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Ebool_2ETYPE__DEFINITION @ A_27a @ A_27b )
= ( ^ [V0P: A_27a > $o,V1rep: A_27b > A_27a] :
( c_2Ebool_2E_2F_5C
@ ( c_2Ebool_2E_21 @ A_27b
@ ^ [V2x_27: A_27b] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V3x_27_27: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D @ A_27a @ ( V1rep @ V2x_27 ) @ ( V1rep @ V3x_27_27 ) ) @ ( c_2Emin_2E_3D @ A_27b @ V2x_27 @ V3x_27_27 ) ) ) )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V4x: A_27a] :
( c_2Emin_2E_3D @ $o @ ( V0P @ V4x )
@ ( c_2Ebool_2E_3F @ A_27b
@ ^ [V5x_27: A_27b] : ( c_2Emin_2E_3D @ A_27a @ V4x @ ( V1rep @ V5x_27 ) ) ) ) ) ) ) ) ).
thf(thm_2Ebool_2Eliteral__case__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b )
= ( ^ [V0f: A_27a > A_27b,V1x: A_27a] : ( V0f @ V1x ) ) ) ).
thf(thm_2Ebool_2EIN__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Ebool_2EIN @ A_27a )
= ( ^ [V0x: A_27a,V1f: A_27a > $o] : ( V1f @ V0x ) ) ) ).
thf(thm_2Ebool_2ERES__FORALL__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Ebool_2ERES__FORALL @ A_27a )
= ( ^ [V0p: A_27a > $o,V1m: A_27a > $o] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V2x: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0p ) @ ( V1m @ V2x ) ) ) ) ) ).
thf(thm_2Ebool_2ERES__EXISTS__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a )
= ( ^ [V0p: A_27a > $o,V1m: A_27a > $o] :
( c_2Ebool_2E_3F @ A_27a
@ ^ [V2x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0p ) @ ( V1m @ V2x ) ) ) ) ) ).
thf(thm_2Ebool_2ERES__EXISTS__UNIQUE__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a )
= ( ^ [V0p: A_27a > $o,V1m: A_27a > $o] :
( c_2Ebool_2E_2F_5C
@ ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0p
@ ^ [V2x: A_27a] : ( V1m @ V2x ) )
@ ( c_2Ebool_2ERES__FORALL @ A_27a @ V0p
@ ^ [V3x: A_27a] :
( c_2Ebool_2ERES__FORALL @ A_27a @ V0p
@ ^ [V4y: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( V1m @ V3x ) @ ( V1m @ V4y ) ) @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V4y ) ) ) ) ) ) ) ).
thf(thm_2Ebool_2ERES__SELECT__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Ebool_2ERES__SELECT @ A_27a )
= ( ^ [V0p: A_27a > $o,V1m: A_27a > $o] :
( c_2Emin_2E_40 @ A_27a
@ ^ [V2x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0p ) @ ( V1m @ V2x ) ) ) ) ) ).
thf(thm_2Ebool_2EBOUNDED__DEF,axiom,
( c_2Ebool_2EBOUNDED
= ( ^ [V0v: $o] : c_2Ebool_2ET ) ) ).
thf(thm_2Ebool_2EDATATYPE__TAG__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Ebool_2EDATATYPE @ A_27a )
= ( ^ [V0x: A_27a] : c_2Ebool_2ET ) ) ).
thf(thm_2Ebool_2ERES__ABSTRACT__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0p: A_27a > $o,V1m: A_27a > A_27b,V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0p )
=> ( ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V0p @ V1m @ V2x )
= ( V1m @ V2x ) ) )
& ! [V3p: A_27a > $o,V4m1: A_27a > A_27b,V5m2: A_27a > A_27b] :
( ! [V6x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V6x @ V3p )
=> ( ( V4m1 @ V6x )
= ( V5m2 @ V6x ) ) )
=> ( ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V3p @ V4m1 )
= ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V3p @ V5m2 ) ) ) ) ).
thf(thm_2Ebool_2Eitself__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Ebool_2Eitself @ A_27a ) > A_27a] : ( c_2Ebool_2ETYPE__DEFINITION @ A_27a @ ( tyop_2Ebool_2Eitself @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a @ ( c_2Ebool_2EARB @ A_27a ) ) @ V0rep ) ).
thf(thm_2Ebool_2Eitself__case__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0b: A_27b] :
( ( c_2Ebool_2Eitself__case @ A_27a @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27a ) @ V0b )
= V0b ) ).
thf(thm_2Ebool_2ETRUTH,axiom,
c_2Ebool_2ET ).
thf(thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1: $o,V1t2: $o] :
( ( V0t1
=> V1t2 )
=> ( ( V1t2
=> V0t1 )
=> ( V0t1 = V1t2 ) ) ) ).
thf(thm_2Ebool_2EFALSITY,axiom,
! [V0t: $o] :
( c_2Ebool_2EF
=> V0t ) ).
thf(thm_2Ebool_2EETA__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0M: A_27a > A_27b] :
( ( ^ [V1x: A_27a] : ( V0M @ V1x ) )
= V0M ) ).
thf(thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
! [V0t: $o] :
( V0t
| ( (~) @ V0t ) ) ).
thf(thm_2Ebool_2EBETA__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1y: A_27a] :
( ( ^ [V2x: A_27a] : ( V0f @ V2x )
@ V1y )
= ( V0f @ V1y ) ) ).
thf(thm_2Ebool_2ELET__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1x: A_27a] :
( ( c_2Ebool_2ELET @ A_27a @ A_27b @ V0f @ V1x )
= ( V0f @ V1x ) ) ).
thf(thm_2Ebool_2EFORALL__THM,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Ebool_2EEXISTS__THM,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Ebool_2EABS__SIMP,axiom,
! [A_27a: $tType,A_27b: $tType,V0t1: A_27a,V1t2: A_27b] :
( ( ^ [V2x: A_27b] : V0t1
@ V1t2 )
= V0t1 ) ).
thf(thm_2Ebool_2EFORALL__SIMP,axiom,
! [A_27a: $tType,V0t: $o] :
( ! [V1x: A_27a] : V0t
<=> V0t ) ).
thf(thm_2Ebool_2EEXISTS__SIMP,axiom,
! [A_27a: $tType,V0t: $o] :
( ? [V1x: A_27a] : V0t
<=> V0t ) ).
thf(thm_2Ebool_2EAND__INTRO__THM,axiom,
! [V0t1: $o,V1t2: $o] :
( V0t1
=> ( V1t2
=> ( V0t1
& V1t2 ) ) ) ).
thf(thm_2Ebool_2EAND1__THM,axiom,
! [V0t1: $o,V1t2: $o] :
( ( V0t1
& V1t2 )
=> V0t1 ) ).
thf(thm_2Ebool_2EAND2__THM,axiom,
! [V0t1: $o,V1t2: $o] :
( ( V0t1
& V1t2 )
=> V1t2 ) ).
thf(thm_2Ebool_2ECONJ__SYM,axiom,
! [V0t1: $o,V1t2: $o] :
( ( V0t1
& V1t2 )
<=> ( V1t2
& V0t1 ) ) ).
thf(thm_2Ebool_2ECONJ__COMM,axiom,
! [V0t1: $o,V1t2: $o] :
( ( V0t1
& V1t2 )
<=> ( V1t2
& V0t1 ) ) ).
thf(thm_2Ebool_2ECONJ__ASSOC,axiom,
! [V0t1: $o,V1t2: $o,V2t3: $o] :
( ( V0t1
& V1t2
& V2t3 )
<=> ( V0t1
& V1t2
& V2t3 ) ) ).
thf(thm_2Ebool_2EOR__INTRO__THM1,axiom,
! [V0t1: $o,V1t2: $o] :
( V0t1
=> ( V0t1
| V1t2 ) ) ).
thf(thm_2Ebool_2EOR__INTRO__THM2,axiom,
! [V0t1: $o,V1t2: $o] :
( V1t2
=> ( V0t1
| V1t2 ) ) ).
thf(thm_2Ebool_2EOR__ELIM__THM,axiom,
! [V0t: $o,V1t1: $o,V2t2: $o] :
( ( V1t1
| V2t2 )
=> ( ( V1t1
=> V0t )
=> ( ( V2t2
=> V0t )
=> V0t ) ) ) ).
thf(thm_2Ebool_2EIMP__F,axiom,
! [V0t: $o] :
( ( V0t
=> c_2Ebool_2EF )
=> ( (~) @ V0t ) ) ).
thf(thm_2Ebool_2EF__IMP,axiom,
! [V0t: $o] :
( ( (~) @ V0t )
=> ( V0t
=> c_2Ebool_2EF ) ) ).
thf(thm_2Ebool_2ENOT__F,axiom,
! [V0t: $o] :
( ( (~) @ V0t )
=> ( V0t = c_2Ebool_2EF ) ) ).
thf(thm_2Ebool_2ENOT__AND,axiom,
! [V0t: $o] :
( (~)
@ ( V0t
& ( (~) @ V0t ) ) ) ).
thf(thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET
& V0t )
<=> V0t )
& ( ( V0t
& c_2Ebool_2ET )
<=> V0t )
& ( ( c_2Ebool_2EF
& V0t )
<=> c_2Ebool_2EF )
& ( ( V0t
& c_2Ebool_2EF )
<=> c_2Ebool_2EF )
& ( ( V0t
& V0t )
<=> V0t ) ) ).
thf(thm_2Ebool_2EOR__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET
| V0t )
<=> c_2Ebool_2ET )
& ( ( V0t
| c_2Ebool_2ET )
<=> c_2Ebool_2ET )
& ( ( c_2Ebool_2EF
| V0t )
<=> V0t )
& ( ( V0t
| c_2Ebool_2EF )
<=> V0t )
& ( ( V0t
| V0t )
<=> V0t ) ) ).
thf(thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET
=> V0t )
<=> V0t )
& ( ( V0t
=> c_2Ebool_2ET )
<=> c_2Ebool_2ET )
& ( ( c_2Ebool_2EF
=> V0t )
<=> c_2Ebool_2ET )
& ( ( V0t
=> V0t )
<=> c_2Ebool_2ET )
& ( ( V0t
=> c_2Ebool_2EF )
<=> ( (~) @ V0t ) ) ) ).
thf(thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: $o] :
( ( (~) @ ( (~) @ V0t ) )
<=> V0t )
& ( ( (~) @ c_2Ebool_2ET )
<=> c_2Ebool_2EF )
& ( ( (~) @ c_2Ebool_2EF )
<=> c_2Ebool_2ET ) ) ).
thf(thm_2Ebool_2EEQ__REFL,axiom,
! [A_27a: $tType,V0x: A_27a] : ( V0x = V0x ) ).
thf(thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( V0x = V0x )
<=> c_2Ebool_2ET ) ).
thf(thm_2Ebool_2EEQ__SYM,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
( ( V0x = V1y )
=> ( V1y = V0x ) ) ).
thf(thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
( ( V0x = V1y )
<=> ( V1y = V0x ) ) ).
thf(thm_2Ebool_2EEQ__EXT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > A_27b] :
( ! [V2x: A_27a] :
( ( V0f @ V2x )
= ( V1g @ V2x ) )
=> ( V0f = V1g ) ) ).
thf(thm_2Ebool_2EFUN__EQ__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > A_27b] :
( ( V0f = V1g )
<=> ! [V2x: A_27a] :
( ( V0f @ V2x )
= ( V1g @ V2x ) ) ) ).
thf(thm_2Ebool_2EEQ__TRANS,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a,V2z: A_27a] :
( ( ( V0x = V1y )
& ( V1y = V2z ) )
=> ( V0x = V2z ) ) ).
thf(thm_2Ebool_2EBOOL__EQ__DISTINCT,axiom,
( ( (~) @ ( c_2Ebool_2ET = c_2Ebool_2EF ) )
& ( (~) @ ( c_2Ebool_2EF = c_2Ebool_2ET ) ) ) ).
thf(thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET = V0t )
<=> V0t )
& ( ( V0t = c_2Ebool_2ET )
<=> V0t )
& ( ( c_2Ebool_2EF = V0t )
<=> ( (~) @ V0t ) )
& ( ( V0t = c_2Ebool_2EF )
<=> ( (~) @ V0t ) ) ) ).
thf(thm_2Ebool_2ECOND__CLAUSES,axiom,
! [A_27a: $tType,V0t1: A_27a,V1t2: A_27a] :
( ( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2ET @ V0t1 @ V1t2 )
= V0t1 )
& ( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2EF @ V0t1 @ V1t2 )
= V1t2 ) ) ).
thf(thm_2Ebool_2ECOND__ID,axiom,
! [A_27a: $tType,V0b: $o,V1t: A_27a] :
( ( c_2Ebool_2ECOND @ A_27a @ V0b @ V1t @ V1t )
= V1t ) ).
thf(thm_2Ebool_2ESELECT__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( V0P
@ ( c_2Emin_2E_40 @ A_27a
@ ^ [V1x: A_27a] : ( V0P @ V1x ) ) )
<=> ? [V2x: A_27a] : ( V0P @ V2x ) ) ).
thf(thm_2Ebool_2ESELECT__REFL,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Emin_2E_40 @ A_27a
@ ^ [V1y: A_27a] : ( c_2Emin_2E_3D @ A_27a @ V1y @ V0x ) )
= V0x ) ).
thf(thm_2Ebool_2ESELECT__REFL__2,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Emin_2E_40 @ A_27a
@ ^ [V1y: A_27a] : ( c_2Emin_2E_3D @ A_27a @ V0x @ V1y ) )
= V0x ) ).
thf(thm_2Ebool_2ESELECT__UNIQUE,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1x: A_27a] :
( ! [V2y: A_27a] :
( ( V0P @ V2y )
<=> ( V2y = V1x ) )
=> ( ( c_2Emin_2E_40 @ A_27a @ V0P )
= V1x ) ) ).
thf(thm_2Ebool_2ESELECT__ELIM__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
( ( ? [V2x: A_27a] : ( V0P @ V2x )
& ! [V3x: A_27a] :
( ( V0P @ V3x )
=> ( V1Q @ V3x ) ) )
=> ( V1Q @ ( c_2Emin_2E_40 @ A_27a @ V0P ) ) ) ).
thf(thm_2Ebool_2ENOT__FORALL__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( (~)
@ ! [V1x: A_27a] : ( V0P @ V1x ) )
<=> ? [V2x: A_27a] : ( (~) @ ( V0P @ V2x ) ) ) ).
thf(thm_2Ebool_2ENOT__EXISTS__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( (~)
@ ? [V1x: A_27a] : ( V0P @ V1x ) )
<=> ! [V2x: A_27a] : ( (~) @ ( V0P @ V2x ) ) ) ).
thf(thm_2Ebool_2EFORALL__AND__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
( ! [V2x: A_27a] :
( ( V0P @ V2x )
& ( V1Q @ V2x ) )
<=> ( ! [V3x: A_27a] : ( V0P @ V3x )
& ! [V4x: A_27a] : ( V1Q @ V4x ) ) ) ).
thf(thm_2Ebool_2ELEFT__AND__FORALL__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
( ( ! [V2x: A_27a] : ( V0P @ V2x )
& V1Q )
<=> ! [V3x: A_27a] :
( ( V0P @ V3x )
& V1Q ) ) ).
thf(thm_2Ebool_2ERIGHT__AND__FORALL__THM,axiom,
! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
( ( V0P
& ! [V2x: A_27a] : ( V1Q @ V2x ) )
<=> ! [V3x: A_27a] :
( V0P
& ( V1Q @ V3x ) ) ) ).
thf(thm_2Ebool_2EEXISTS__OR__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
( ? [V2x: A_27a] :
( ( V0P @ V2x )
| ( V1Q @ V2x ) )
<=> ( ? [V3x: A_27a] : ( V0P @ V3x )
| ? [V4x: A_27a] : ( V1Q @ V4x ) ) ) ).
thf(thm_2Ebool_2ELEFT__OR__EXISTS__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
( ( ? [V2x: A_27a] : ( V0P @ V2x )
| V1Q )
<=> ? [V3x: A_27a] :
( ( V0P @ V3x )
| V1Q ) ) ).
thf(thm_2Ebool_2ERIGHT__OR__EXISTS__THM,axiom,
! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
( ( V0P
| ? [V2x: A_27a] : ( V1Q @ V2x ) )
<=> ? [V3x: A_27a] :
( V0P
| ( V1Q @ V3x ) ) ) ).
thf(thm_2Ebool_2EBOTH__EXISTS__AND__THM,axiom,
! [A_27a: $tType,V0P: $o,V1Q: $o] :
( ? [V2x: A_27a] :
( V0P
& V1Q )
<=> ( ? [V3x: A_27a] : V0P
& ? [V4x: A_27a] : V1Q ) ) ).
thf(thm_2Ebool_2ELEFT__EXISTS__AND__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
( ? [V2x: A_27a] :
( ( V0P @ V2x )
& V1Q )
<=> ( ? [V3x: A_27a] : ( V0P @ V3x )
& V1Q ) ) ).
thf(thm_2Ebool_2ERIGHT__EXISTS__AND__THM,axiom,
! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
( ? [V2x: A_27a] :
( V0P
& ( V1Q @ V2x ) )
<=> ( V0P
& ? [V3x: A_27a] : ( V1Q @ V3x ) ) ) ).
thf(thm_2Ebool_2EBOTH__FORALL__OR__THM,axiom,
! [A_27a: $tType,V0P: $o,V1Q: $o] :
( ! [V2x: A_27a] :
( V0P
| V1Q )
<=> ( ! [V3x: A_27a] : V0P
| ! [V4x: A_27a] : V1Q ) ) ).
thf(thm_2Ebool_2ELEFT__FORALL__OR__THM,axiom,
! [A_27a: $tType,V0Q: $o,V1P: A_27a > $o] :
( ! [V2x: A_27a] :
( ( V1P @ V2x )
| V0Q )
<=> ( ! [V3x: A_27a] : ( V1P @ V3x )
| V0Q ) ) ).
thf(thm_2Ebool_2ERIGHT__FORALL__OR__THM,axiom,
! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
( ! [V2x: A_27a] :
( V0P
| ( V1Q @ V2x ) )
<=> ( V0P
| ! [V3x: A_27a] : ( V1Q @ V3x ) ) ) ).
thf(thm_2Ebool_2EBOTH__FORALL__IMP__THM,axiom,
! [A_27a: $tType,V0P: $o,V1Q: $o] :
( ! [V2x: A_27a] :
( V0P
=> V1Q )
<=> ( ? [V3x: A_27a] : V0P
=> ! [V4x: A_27a] : V1Q ) ) ).
thf(thm_2Ebool_2ELEFT__FORALL__IMP__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
( ! [V2x: A_27a] :
( ( V0P @ V2x )
=> V1Q )
<=> ( ? [V3x: A_27a] : ( V0P @ V3x )
=> V1Q ) ) ).
thf(thm_2Ebool_2ERIGHT__FORALL__IMP__THM,axiom,
! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
( ! [V2x: A_27a] :
( V0P
=> ( V1Q @ V2x ) )
<=> ( V0P
=> ! [V3x: A_27a] : ( V1Q @ V3x ) ) ) ).
thf(thm_2Ebool_2EBOTH__EXISTS__IMP__THM,axiom,
! [A_27a: $tType,V0P: $o,V1Q: $o] :
( ? [V2x: A_27a] :
( V0P
=> V1Q )
<=> ( ! [V3x: A_27a] : V0P
=> ? [V4x: A_27a] : V1Q ) ) ).
thf(thm_2Ebool_2ELEFT__EXISTS__IMP__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
( ? [V2x: A_27a] :
( ( V0P @ V2x )
=> V1Q )
<=> ( ! [V3x: A_27a] : ( V0P @ V3x )
=> V1Q ) ) ).
thf(thm_2Ebool_2ERIGHT__EXISTS__IMP__THM,axiom,
! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
( ? [V2x: A_27a] :
( V0P
=> ( V1Q @ V2x ) )
<=> ( V0P
=> ? [V3x: A_27a] : ( V1Q @ V3x ) ) ) ).
thf(thm_2Ebool_2EOR__IMP__THM,axiom,
! [V0A: $o,V1B: $o] :
( ( V0A
<=> ( V1B
| V0A ) )
<=> ( V1B
=> V0A ) ) ).
thf(thm_2Ebool_2ENOT__IMP,axiom,
! [V0A: $o,V1B: $o] :
( ( (~)
@ ( V0A
=> V1B ) )
<=> ( V0A
& ( (~) @ V1B ) ) ) ).
thf(thm_2Ebool_2EDISJ__ASSOC,axiom,
! [V0A: $o,V1B: $o,V2C: $o] :
( ( V0A
| V1B
| V2C )
<=> ( V0A
| V1B
| V2C ) ) ).
thf(thm_2Ebool_2EDISJ__SYM,axiom,
! [V0A: $o,V1B: $o] :
( ( V0A
| V1B )
<=> ( V1B
| V0A ) ) ).
thf(thm_2Ebool_2EDISJ__COMM,axiom,
! [V0A: $o,V1B: $o] :
( ( V0A
| V1B )
<=> ( V1B
| V0A ) ) ).
thf(thm_2Ebool_2EDE__MORGAN__THM,axiom,
! [V0A: $o,V1B: $o] :
( ( ( (~)
@ ( V0A
& V1B ) )
<=> ( ( (~) @ V0A )
| ( (~) @ V1B ) ) )
& ( ( (~)
@ ( V0A
| V1B ) )
<=> ( ( (~) @ V0A )
& ( (~) @ V1B ) ) ) ) ).
thf(thm_2Ebool_2ELEFT__AND__OVER__OR,axiom,
! [V0A: $o,V1B: $o,V2C: $o] :
( ( V0A
& ( V1B
| V2C ) )
<=> ( ( V0A
& V1B )
| ( V0A
& V2C ) ) ) ).
thf(thm_2Ebool_2ERIGHT__AND__OVER__OR,axiom,
! [V0A: $o,V1B: $o,V2C: $o] :
( ( ( V1B
| V2C )
& V0A )
<=> ( ( V1B
& V0A )
| ( V2C
& V0A ) ) ) ).
thf(thm_2Ebool_2ELEFT__OR__OVER__AND,axiom,
! [V0A: $o,V1B: $o,V2C: $o] :
( ( V0A
| ( V1B
& V2C ) )
<=> ( ( V0A
| V1B )
& ( V0A
| V2C ) ) ) ).
thf(thm_2Ebool_2ERIGHT__OR__OVER__AND,axiom,
! [V0A: $o,V1B: $o,V2C: $o] :
( ( ( V1B
& V2C )
| V0A )
<=> ( ( V1B
| V0A )
& ( V2C
| V0A ) ) ) ).
thf(thm_2Ebool_2EIMP__DISJ__THM,axiom,
! [V0A: $o,V1B: $o] :
( ( V0A
=> V1B )
<=> ( ( (~) @ V0A )
| V1B ) ) ).
thf(thm_2Ebool_2EDISJ__IMP__THM,axiom,
! [V0P: $o,V1Q: $o,V2R: $o] :
( ( ( V0P
| V1Q )
=> V2R )
<=> ( ( V0P
=> V2R )
& ( V1Q
=> V2R ) ) ) ).
thf(thm_2Ebool_2EIMP__CONJ__THM,axiom,
! [V0P: $o,V1Q: $o,V2R: $o] :
( ( V0P
=> ( V1Q
& V2R ) )
<=> ( ( V0P
=> V1Q )
& ( V0P
=> V2R ) ) ) ).
thf(thm_2Ebool_2EIMP__F__EQ__F,axiom,
! [V0t: $o] :
( ( V0t
=> c_2Ebool_2EF )
<=> ( V0t = c_2Ebool_2EF ) ) ).
thf(thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1: $o,V1t2: $o,V2t3: $o] :
( ( V0t1
=> ( V1t2
=> V2t3 ) )
<=> ( ( V0t1
& V1t2 )
=> V2t3 ) ) ).
thf(thm_2Ebool_2EEQ__IMP__THM,axiom,
! [V0t1: $o,V1t2: $o] :
( ( V0t1 = V1t2 )
<=> ( ( V0t1
=> V1t2 )
& ( V1t2
=> V0t1 ) ) ) ).
thf(thm_2Ebool_2EEQ__EXPAND,axiom,
! [V0t1: $o,V1t2: $o] :
( ( V0t1 = V1t2 )
<=> ( ( V0t1
& V1t2 )
| ( ( (~) @ V0t1 )
& ( (~) @ V1t2 ) ) ) ) ).
thf(thm_2Ebool_2ECOND__RATOR,axiom,
! [A_27a: $tType,A_27b: $tType,V0b: $o,V1f: A_27a > A_27b,V2g: A_27a > A_27b,V3x: A_27a] :
( ( c_2Ebool_2ECOND @ ( A_27a > A_27b ) @ V0b @ V1f @ V2g @ V3x )
= ( c_2Ebool_2ECOND @ A_27b @ V0b @ ( V1f @ V3x ) @ ( V2g @ V3x ) ) ) ).
thf(thm_2Ebool_2ECOND__RAND,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1b: $o,V2x: A_27a,V3y: A_27a] :
( ( V0f @ ( c_2Ebool_2ECOND @ A_27a @ V1b @ V2x @ V3y ) )
= ( c_2Ebool_2ECOND @ A_27b @ V1b @ ( V0f @ V2x ) @ ( V0f @ V3y ) ) ) ).
thf(thm_2Ebool_2ECOND__ABS,axiom,
! [A_27a: $tType,A_27b: $tType,V0b: $o,V1f: A_27a > A_27b,V2g: A_27a > A_27b] :
( ( ^ [V3x: A_27a] : ( c_2Ebool_2ECOND @ A_27b @ V0b @ ( V1f @ V3x ) @ ( V2g @ V3x ) ) )
= ( c_2Ebool_2ECOND @ ( A_27a > A_27b ) @ V0b @ V1f @ V2g ) ) ).
thf(thm_2Ebool_2ECOND__EXPAND,axiom,
! [V0b: $o,V1t1: $o,V2t2: $o] :
( ( c_2Ebool_2ECOND @ $o @ V0b @ V1t1 @ V2t2 )
<=> ( ( ( (~) @ V0b )
| V1t1 )
& ( V0b
| V2t2 ) ) ) ).
thf(thm_2Ebool_2ECOND__EXPAND__IMP,axiom,
! [V0b: $o,V1t1: $o,V2t2: $o] :
( ( c_2Ebool_2ECOND @ $o @ V0b @ V1t1 @ V2t2 )
<=> ( ( V0b
=> V1t1 )
& ( ( (~) @ V0b )
=> V2t2 ) ) ) ).
thf(thm_2Ebool_2ECOND__EXPAND__OR,axiom,
! [V0b: $o,V1t1: $o,V2t2: $o] :
( ( c_2Ebool_2ECOND @ $o @ V0b @ V1t1 @ V2t2 )
<=> ( ( V0b
& V1t1 )
| ( ( (~) @ V0b )
& V2t2 ) ) ) ).
thf(thm_2Ebool_2ETYPE__DEFINITION__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1rep: A_27b > A_27a] :
( ( c_2Ebool_2ETYPE__DEFINITION @ A_27a @ A_27b @ V0P @ V1rep )
<=> ( ! [V2x_27: A_27b,V3x_27_27: A_27b] :
( ( ( V1rep @ V2x_27 )
= ( V1rep @ V3x_27_27 ) )
=> ( V2x_27 = V3x_27_27 ) )
& ! [V4x: A_27a] :
( ( V0P @ V4x )
<=> ? [V5x_27: A_27b] :
( V4x
= ( V1rep @ V5x_27 ) ) ) ) ) ).
thf(thm_2Ebool_2EONTO__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ( c_2Ebool_2EONTO @ A_27a @ A_27b @ V0f )
<=> ! [V1y: A_27b] :
? [V2x: A_27a] :
( V1y
= ( V0f @ V2x ) ) ) ).
thf(thm_2Ebool_2EONE__ONE__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ( c_2Ebool_2EONE__ONE @ A_27a @ A_27b @ V0f )
<=> ! [V1x1: A_27a,V2x2: A_27a] :
( ( ( V0f @ V1x1 )
= ( V0f @ V2x2 ) )
=> ( V1x1 = V2x2 ) ) ) ).
thf(thm_2Ebool_2EABS__REP__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o] :
( ? [V1rep: A_27b > A_27a] : ( c_2Ebool_2ETYPE__DEFINITION @ A_27a @ A_27b @ V0P @ V1rep )
=> ? [V2rep: A_27b > A_27a,V3abs: A_27a > A_27b] :
( ! [V4a: A_27b] :
( ( V3abs @ ( V2rep @ V4a ) )
= V4a )
& ! [V5r: A_27a] :
( ( V0P @ V5r )
<=> ( ( V2rep @ ( V3abs @ V5r ) )
= V5r ) ) ) ) ).
thf(thm_2Ebool_2ELET__RAND,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27b > $o,V1N: A_27a > A_27b,V2M: A_27a] :
( ( V0P
@ ( c_2Ebool_2ELET @ A_27a @ A_27b
@ ^ [V3x: A_27a] : ( V1N @ V3x )
@ V2M ) )
= ( c_2Ebool_2ELET @ A_27a @ $o
@ ^ [V4x: A_27a] : ( V0P @ ( V1N @ V4x ) )
@ V2M ) ) ).
thf(thm_2Ebool_2ELET__RATOR,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0b: A_27b,V1N: A_27a > A_27b > A_27c,V2M: A_27a] :
( ( c_2Ebool_2ELET @ A_27a @ ( A_27b > A_27c )
@ ^ [V3x: A_27a] : ( V1N @ V3x )
@ V2M
@ V0b )
= ( c_2Ebool_2ELET @ A_27a @ A_27c
@ ^ [V4x: A_27a] : ( V1N @ V4x @ V0b )
@ V2M ) ) ).
thf(thm_2Ebool_2ESWAP__FORALL__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
( ! [V1x: A_27a,V2y: A_27b] : ( V0P @ V1x @ V2y )
<=> ! [V3y: A_27b,V4x: A_27a] : ( V0P @ V4x @ V3y ) ) ).
thf(thm_2Ebool_2ESWAP__EXISTS__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
( ? [V1x: A_27a,V2y: A_27b] : ( V0P @ V1x @ V2y )
<=> ? [V3y: A_27b,V4x: A_27a] : ( V0P @ V4x @ V3y ) ) ).
thf(thm_2Ebool_2EEXISTS__UNIQUE__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V1x: A_27a] : ( V0P @ V1x ) )
<=> ( ? [V2x: A_27a] : ( V0P @ V2x )
& ! [V3x: A_27a,V4y: A_27a] :
( ( ( V0P @ V3x )
& ( V0P @ V4y ) )
=> ( V3x = V4y ) ) ) ) ).
thf(thm_2Ebool_2ELET__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > A_27b,V2M: A_27a,V3N: A_27a] :
( ( ( V2M = V3N )
& ! [V4x: A_27a] :
( ( V4x = V3N )
=> ( ( V0f @ V4x )
= ( V1g @ V4x ) ) ) )
=> ( ( c_2Ebool_2ELET @ A_27a @ A_27b @ V0f @ V2M )
= ( c_2Ebool_2ELET @ A_27a @ A_27b @ V1g @ V3N ) ) ) ).
thf(thm_2Ebool_2EIMP__CONG,axiom,
! [V0x: $o,V1x_27: $o,V2y: $o,V3y_27: $o] :
( ( ( V0x = V1x_27 )
& ( V1x_27
=> ( V2y = V3y_27 ) ) )
=> ( ( V0x
=> V2y )
<=> ( V1x_27
=> V3y_27 ) ) ) ).
thf(thm_2Ebool_2EAND__CONG,axiom,
! [V0P: $o,V1P_27: $o,V2Q: $o,V3Q_27: $o] :
( ( ( V2Q
=> ( V0P = V1P_27 ) )
& ( V1P_27
=> ( V2Q = V3Q_27 ) ) )
=> ( ( V0P
& V2Q )
<=> ( V1P_27
& V3Q_27 ) ) ) ).
thf(thm_2Ebool_2ELEFT__AND__CONG,axiom,
! [V0P: $o,V1P_27: $o,V2Q: $o,V3Q_27: $o] :
( ( ( V0P = V1P_27 )
& ( V1P_27
=> ( V2Q = V3Q_27 ) ) )
=> ( ( V0P
& V2Q )
<=> ( V1P_27
& V3Q_27 ) ) ) ).
thf(thm_2Ebool_2EOR__CONG,axiom,
! [V0P: $o,V1P_27: $o,V2Q: $o,V3Q_27: $o] :
( ( ( ( (~) @ V2Q )
=> ( V0P = V1P_27 ) )
& ( ( (~) @ V1P_27 )
=> ( V2Q = V3Q_27 ) ) )
=> ( ( V0P
| V2Q )
<=> ( V1P_27
| V3Q_27 ) ) ) ).
thf(thm_2Ebool_2ELEFT__OR__CONG,axiom,
! [V0P: $o,V1P_27: $o,V2Q: $o,V3Q_27: $o] :
( ( ( V0P = V1P_27 )
& ( ( (~) @ V1P_27 )
=> ( V2Q = V3Q_27 ) ) )
=> ( ( V0P
| V2Q )
<=> ( V1P_27
| V3Q_27 ) ) ) ).
thf(thm_2Ebool_2ECOND__CONG,axiom,
! [A_27a: $tType,V0P: $o,V1Q: $o,V2x: A_27a,V3x_27: A_27a,V4y: A_27a,V5y_27: A_27a] :
( ( ( V0P = V1Q )
& ( V1Q
=> ( V2x = V3x_27 ) )
& ( ( (~) @ V1Q )
=> ( V4y = V5y_27 ) ) )
=> ( ( c_2Ebool_2ECOND @ A_27a @ V0P @ V2x @ V4y )
= ( c_2Ebool_2ECOND @ A_27a @ V1Q @ V3x_27 @ V5y_27 ) ) ) ).
thf(thm_2Ebool_2ERES__FORALL__CONG,axiom,
! [A_27a: $tType,V0g: A_27a > $o,V1f: A_27a > $o,V2Q: A_27a > $o,V3P: A_27a > $o] :
( ( V3P = V2Q )
=> ( ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2Q )
=> ( ( V1f @ V4x )
= ( V0g @ V4x ) ) )
=> ( ( c_2Ebool_2ERES__FORALL @ A_27a @ V3P @ V1f )
= ( c_2Ebool_2ERES__FORALL @ A_27a @ V2Q @ V0g ) ) ) ) ).
thf(thm_2Ebool_2ERES__EXISTS__CONG,axiom,
! [A_27a: $tType,V0g: A_27a > $o,V1f: A_27a > $o,V2Q: A_27a > $o,V3P: A_27a > $o] :
( ( V3P = V2Q )
=> ( ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2Q )
=> ( ( V1f @ V4x )
= ( V0g @ V4x ) ) )
=> ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V3P @ V1f )
= ( c_2Ebool_2ERES__EXISTS @ A_27a @ V2Q @ V0g ) ) ) ) ).
thf(thm_2Ebool_2EMONO__AND,axiom,
! [V0z: $o,V1y: $o,V2x: $o,V3w: $o] :
( ( ( V2x
=> V1y )
& ( V0z
=> V3w ) )
=> ( ( V2x
& V0z )
=> ( V1y
& V3w ) ) ) ).
thf(thm_2Ebool_2EMONO__OR,axiom,
! [V0z: $o,V1y: $o,V2x: $o,V3w: $o] :
( ( ( V2x
=> V1y )
& ( V0z
=> V3w ) )
=> ( ( V2x
| V0z )
=> ( V1y
| V3w ) ) ) ).
thf(thm_2Ebool_2EMONO__IMP,axiom,
! [V0z: $o,V1y: $o,V2x: $o,V3w: $o] :
( ( ( V1y
=> V2x )
& ( V0z
=> V3w ) )
=> ( ( V2x
=> V0z )
=> ( V1y
=> V3w ) ) ) ).
thf(thm_2Ebool_2EMONO__NOT,axiom,
! [V0y: $o,V1x: $o] :
( ( V0y
=> V1x )
=> ( ( (~) @ V1x )
=> ( (~) @ V0y ) ) ) ).
thf(thm_2Ebool_2EMONO__NOT__EQ,axiom,
! [V0y: $o,V1x: $o] :
( ( V0y
=> V1x )
<=> ( ( (~) @ V1x )
=> ( (~) @ V0y ) ) ) ).
thf(thm_2Ebool_2EMONO__ALL,axiom,
! [A_27a: $tType,V0Q: A_27a > $o,V1P: A_27a > $o] :
( ! [V2x: A_27a] :
( ( V1P @ V2x )
=> ( V0Q @ V2x ) )
=> ( ! [V3x: A_27a] : ( V1P @ V3x )
=> ! [V4x: A_27a] : ( V0Q @ V4x ) ) ) ).
thf(thm_2Ebool_2EMONO__EXISTS,axiom,
! [A_27a: $tType,V0Q: A_27a > $o,V1P: A_27a > $o] :
( ! [V2x: A_27a] :
( ( V1P @ V2x )
=> ( V0Q @ V2x ) )
=> ( ? [V3x: A_27a] : ( V1P @ V3x )
=> ? [V4x: A_27a] : ( V0Q @ V4x ) ) ) ).
thf(thm_2Ebool_2EMONO__COND,axiom,
! [V0z: $o,V1y: $o,V2x: $o,V3w: $o,V4b: $o] :
( ( V2x
=> V1y )
=> ( ( V0z
=> V3w )
=> ( ( c_2Ebool_2ECOND @ $o @ V4b @ V2x @ V0z )
=> ( c_2Ebool_2ECOND @ $o @ V4b @ V1y @ V3w ) ) ) ) ).
thf(thm_2Ebool_2EEXISTS__REFL,axiom,
! [A_27a: $tType,V0a: A_27a] :
? [V1x: A_27a] : ( V1x = V0a ) ).
thf(thm_2Ebool_2EEXISTS__UNIQUE__REFL,axiom,
! [A_27a: $tType,V0a: A_27a] :
( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V1x: A_27a] : ( c_2Emin_2E_3D @ A_27a @ V1x @ V0a ) ) ).
thf(thm_2Ebool_2EUNWIND__THM1,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1a: A_27a] :
( ? [V2x: A_27a] :
( ( V1a = V2x )
& ( V0P @ V2x ) )
<=> ( V0P @ V1a ) ) ).
thf(thm_2Ebool_2EUNWIND__THM2,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1a: A_27a] :
( ? [V2x: A_27a] :
( ( V2x = V1a )
& ( V0P @ V2x ) )
<=> ( V0P @ V1a ) ) ).
thf(thm_2Ebool_2EUNWIND__FORALL__THM1,axiom,
! [A_27a: $tType,V0f: A_27a > $o,V1v: A_27a] :
( ! [V2x: A_27a] :
( ( V1v = V2x )
=> ( V0f @ V2x ) )
<=> ( V0f @ V1v ) ) ).
thf(thm_2Ebool_2EUNWIND__FORALL__THM2,axiom,
! [A_27a: $tType,V0f: A_27a > $o,V1v: A_27a] :
( ! [V2x: A_27a] :
( ( V2x = V1v )
=> ( V0f @ V2x ) )
<=> ( V0f @ V1v ) ) ).
thf(thm_2Ebool_2ESKOLEM__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
( ! [V1x: A_27a] :
? [V2y: A_27b] : ( V0P @ V1x @ V2y )
<=> ? [V3f: A_27a > A_27b] :
! [V4x: A_27a] : ( V0P @ V4x @ ( V3f @ V4x ) ) ) ).
thf(thm_2Ebool_2Ebool__case__thm,axiom,
! [A_27a: $tType] :
( ! [V0t1: A_27a,V1t2: A_27a] :
( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2ET @ V0t1 @ V1t2 )
= V0t1 )
& ! [V2t1: A_27a,V3t2: A_27a] :
( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2EF @ V2t1 @ V3t2 )
= V3t2 ) ) ).
thf(thm_2Ebool_2Ebool__case__ID,axiom,
! [A_27a: $tType,V0b: $o,V1t: A_27a] :
( ( c_2Ebool_2ECOND @ A_27a @ V0b @ V1t @ V1t )
= V1t ) ).
thf(thm_2Ebool_2EboolAxiom,axiom,
! [A_27a: $tType,V0t1: A_27a,V1t2: A_27a] :
? [V2fn: $o > A_27a] :
( ( ( V2fn @ c_2Ebool_2ET )
= V0t1 )
& ( ( V2fn @ c_2Ebool_2EF )
= V1t2 ) ) ).
thf(thm_2Ebool_2Ebool__INDUCT,axiom,
! [V0P: $o > $o] :
( ( ( V0P @ c_2Ebool_2ET )
& ( V0P @ c_2Ebool_2EF ) )
=> ! [V1b: $o] : ( V0P @ V1b ) ) ).
thf(thm_2Ebool_2Ebool__case__CONG,axiom,
! [A_27a: $tType,V0P: $o,V1Q: $o,V2x: A_27a,V3x_27: A_27a,V4y: A_27a,V5y_27: A_27a] :
( ( ( V0P = V1Q )
& ( V1Q
=> ( V2x = V3x_27 ) )
& ( ( (~) @ V1Q )
=> ( V4y = V5y_27 ) ) )
=> ( ( c_2Ebool_2ECOND @ A_27a @ V0P @ V2x @ V4y )
= ( c_2Ebool_2ECOND @ A_27a @ V1Q @ V3x_27 @ V5y_27 ) ) ) ).
thf(thm_2Ebool_2EFORALL__BOOL,axiom,
! [V0P: $o > $o] :
( ! [V1b: $o] : ( V0P @ V1b )
<=> ( ( V0P @ c_2Ebool_2ET )
& ( V0P @ c_2Ebool_2EF ) ) ) ).
thf(thm_2Ebool_2EUEXISTS__OR__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
( ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V2x: A_27a] : ( c_2Ebool_2E_5C_2F @ ( V0P @ V2x ) @ ( V1Q @ V2x ) ) )
=> ( ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V3x: A_27a] : ( V0P @ V3x ) )
| ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V4x: A_27a] : ( V1Q @ V4x ) ) ) ) ).
thf(thm_2Ebool_2EUEXISTS__SIMP,axiom,
! [A_27a: $tType,V0t: $o] :
( ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V1x: A_27a] : V0t )
<=> ( V0t
& ! [V2x: A_27a,V3y: A_27a] : ( V2x = V3y ) ) ) ).
thf(thm_2Ebool_2ERES__FORALL__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > $o] :
( ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P @ V1f )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P )
=> ( V1f @ V2x ) ) ) ).
thf(thm_2Ebool_2ERES__EXISTS__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P @ V1f )
<=> ? [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P )
& ( V1f @ V2x ) ) ) ).
thf(thm_2Ebool_2ERES__EXISTS__UNIQUE__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ V0P @ V1f )
<=> ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P
@ ^ [V2x: A_27a] : ( V1f @ V2x ) )
& ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
@ ^ [V3x: A_27a] :
( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
@ ^ [V4y: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( V1f @ V3x ) @ ( V1f @ V4y ) ) @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V4y ) ) ) ) ) ) ).
thf(thm_2Ebool_2ERES__SELECT__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > $o] :
( ( c_2Ebool_2ERES__SELECT @ A_27a @ V0P @ V1f )
= ( c_2Emin_2E_40 @ A_27a
@ ^ [V2x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P ) @ ( V1f @ V2x ) ) ) ) ).
thf(thm_2Ebool_2ERES__FORALL__TRUE,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
@ ^ [V1x: A_27a] : c_2Ebool_2ET )
= c_2Ebool_2ET ) ).
thf(thm_2Ebool_2ERES__EXISTS__FALSE,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P
@ ^ [V1x: A_27a] : c_2Ebool_2EF )
= c_2Ebool_2EF ) ).
thf(thm_2Ebool_2EBOOL__FUN__CASES__THM,axiom,
! [V0f: $o > $o] :
( ( V0f
= ( ^ [V1b: $o] : c_2Ebool_2ET ) )
| ( V0f
= ( ^ [V2b: $o] : c_2Ebool_2EF ) )
| ( V0f
= ( ^ [V3b: $o] : V3b ) )
| ( V0f
= ( ^ [V4b: $o] : ( c_2Ebool_2E_7E @ V4b ) ) ) ) ).
thf(thm_2Ebool_2EBOOL__FUN__INDUCT,axiom,
! [V0P: ( $o > $o ) > $o] :
( ( ( V0P
@ ^ [V1b: $o] : c_2Ebool_2ET )
& ( V0P
@ ^ [V2b: $o] : c_2Ebool_2EF )
& ( V0P
@ ^ [V3b: $o] : V3b )
& ( V0P
@ ^ [V4b: $o] : ( c_2Ebool_2E_7E @ V4b ) ) )
=> ! [V5f: $o > $o] : ( V0P @ V5f ) ) ).
thf(thm_2Ebool_2Eliteral__case__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1x: A_27a] :
( ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b @ V0f @ V1x )
= ( V0f @ V1x ) ) ).
thf(thm_2Ebool_2Eliteral__case__RAND,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: A_27b > A_27c,V1N: A_27a > A_27b,V2M: A_27a] :
( ( V0P
@ ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b
@ ^ [V3x: A_27a] : ( V1N @ V3x )
@ V2M ) )
= ( c_2Ebool_2Eliteral__case @ A_27a @ A_27c
@ ^ [V4x: A_27a] : ( V0P @ ( V1N @ V4x ) )
@ V2M ) ) ).
thf(thm_2Ebool_2Eliteral__case__RATOR,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0b: A_27b,V1N: A_27a > A_27b > A_27c,V2M: A_27a] :
( ( c_2Ebool_2Eliteral__case @ A_27a @ ( A_27b > A_27c )
@ ^ [V3x: A_27a] : ( V1N @ V3x )
@ V2M
@ V0b )
= ( c_2Ebool_2Eliteral__case @ A_27a @ A_27c
@ ^ [V4x: A_27a] : ( V1N @ V4x @ V0b )
@ V2M ) ) ).
thf(thm_2Ebool_2Eliteral__case__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > A_27b,V2M: A_27a,V3N: A_27a] :
( ( ( V2M = V3N )
& ! [V4x: A_27a] :
( ( V4x = V3N )
=> ( ( V0f @ V4x )
= ( V1g @ V4x ) ) ) )
=> ( ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b @ V0f @ V2M )
= ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b @ V1g @ V3N ) ) ) ).
thf(thm_2Ebool_2Eliteral__case__id,axiom,
! [A_27a: $tType,A_27b: $tType,V0u: A_27b,V1t: A_27b,V2a: A_27a] :
( ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b
@ ^ [V3x: A_27a] : ( c_2Ebool_2ECOND @ A_27b @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V2a ) @ V1t @ V0u )
@ V2a )
= V1t ) ).
thf(thm_2Ebool_2EBOUNDED__THM,axiom,
! [V0v: $o] :
( ( c_2Ebool_2EBOUNDED @ V0v )
= c_2Ebool_2ET ) ).
thf(thm_2Ebool_2ELCOMM__THM,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a > A_27a] :
( ! [V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( V0f @ V1x @ ( V0f @ V2y @ V3z ) )
= ( V0f @ ( V0f @ V1x @ V2y ) @ V3z ) )
=> ( ! [V4x: A_27a,V5y: A_27a] :
( ( V0f @ V4x @ V5y )
= ( V0f @ V5y @ V4x ) )
=> ! [V6x: A_27a,V7y: A_27a,V8z: A_27a] :
( ( V0f @ V6x @ ( V0f @ V7y @ V8z ) )
= ( V0f @ V7y @ ( V0f @ V6x @ V8z ) ) ) ) ) ).
thf(thm_2Ebool_2EDATATYPE__TAG__THM,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Ebool_2EDATATYPE @ A_27a @ V0x )
= c_2Ebool_2ET ) ).
thf(thm_2Ebool_2EDATATYPE__BOOL,axiom,
! [V0bool: $o > $o > $o] :
( ( c_2Ebool_2EDATATYPE @ $o @ ( V0bool @ c_2Ebool_2ET @ c_2Ebool_2EF ) )
= c_2Ebool_2ET ) ).
thf(thm_2Ebool_2EITSELF__UNIQUE,axiom,
! [A_27a: $tType,V0i: tyop_2Ebool_2Eitself @ A_27a] :
( V0i
= ( c_2Ebool_2Ethe__value @ A_27a ) ) ).
thf(thm_2Ebool_2Eitself__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,V0e: A_27b] :
? [V1f: ( tyop_2Ebool_2Eitself @ A_27a ) > A_27b] :
( ( V1f @ ( c_2Ebool_2Ethe__value @ A_27a ) )
= V0e ) ).
thf(thm_2Ebool_2Eitself__induction,axiom,
! [A_27a: $tType,V0P: ( tyop_2Ebool_2Eitself @ A_27a ) > $o] :
( ( V0P @ ( c_2Ebool_2Ethe__value @ A_27a ) )
=> ! [V1i: tyop_2Ebool_2Eitself @ A_27a] : ( V0P @ V1i ) ) ).
thf(thm_2Ebool_2EFORALL__itself,axiom,
! [A_27a: $tType,V0P: ( tyop_2Ebool_2Eitself @ A_27a ) > $o] :
( ! [V1x: tyop_2Ebool_2Eitself @ A_27a] : ( V0P @ V1x )
<=> ( V0P @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) ).
thf(thm_2Ebool_2EEXISTS__itself,axiom,
! [A_27a: $tType,V0P: ( tyop_2Ebool_2Eitself @ A_27a ) > $o] :
( ? [V1x: tyop_2Ebool_2Eitself @ A_27a] : ( V0P @ V1x )
<=> ( V0P @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) ).
thf(thm_2Ebool_2EPULL__EXISTS,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
( ( ( ? [V2x: A_27a] : ( V0P @ V2x )
=> V1Q )
<=> ! [V3x: A_27a] :
( ( V0P @ V3x )
=> V1Q ) )
& ( ( ? [V4x: A_27a] : ( V0P @ V4x )
& V1Q )
<=> ? [V5x: A_27a] :
( ( V0P @ V5x )
& V1Q ) )
& ( ( V1Q
& ? [V6x: A_27a] : ( V0P @ V6x ) )
<=> ? [V7x: A_27a] :
( V1Q
& ( V0P @ V7x ) ) ) ) ).
thf(thm_2Ebool_2EPULL__FORALL,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
( ( ( V1Q
=> ! [V2x: A_27a] : ( V0P @ V2x ) )
<=> ! [V3x: A_27a] :
( V1Q
=> ( V0P @ V3x ) ) )
& ( ( ! [V4x: A_27a] : ( V0P @ V4x )
& V1Q )
<=> ! [V5x: A_27a] :
( ( V0P @ V5x )
& V1Q ) )
& ( ( V1Q
& ! [V6x: A_27a] : ( V0P @ V6x ) )
<=> ! [V7x: A_27a] :
( V1Q
& ( V0P @ V7x ) ) ) ) ).
thf(thm_2Ebool_2EPEIRCE,axiom,
! [V0Q: $o,V1P: $o] :
( ( ( V1P
=> V0Q )
=> V1P )
=> V1P ) ).
thf(thm_2Ebool_2EJRH__INDUCT__UTIL,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1t: A_27a] :
( ! [V2x: A_27a] :
( ( V2x = V1t )
=> ( V0P @ V2x ) )
=> ( c_2Ebool_2E_3F @ A_27a @ V0P ) ) ).
thf(thm_2Ebool_2EDISJ__EQ__IMP,axiom,
! [V0A: $o,V1B: $o] :
( ( V0A
| V1B )
<=> ( ( (~) @ V0A )
=> V1B ) ) ).
%------------------------------------------------------------------------------